4.
There is an additional problem in that many of the maps in chaos
studies (e.g., the baker’s transformation) have purely
mathematical origins rather than being derived from some more complex
model for a target system. What connection these maps are supposed to
have with the space of possibilities of actual systems is difficult to
see.

6.
As long as there is some uncertainty in the initial data of a target
system even a very faithful model’s output will diverge away
from the behavior of the target system. This is because any
uncertainty in ascertaining the true initial conditions leads to
divergence in the model behavior from the system behavior (Smith 2003); and, there
is no way to reduce this uncertainty to zero (e.g., Bishop 2003).

7. Some have argued that
classical chaos cannot amplify quantum indeterminacy because of
environmentally induced decoherence (e.g., Berry 2001). However, such
arguments invoking environmental decoherence seriously underestimate
what is going on physically in the relationship between QM and CM
(see §6.3 below).

9.
Control parameters are particular variables or other features of a
system–e.g., temperature, voltage, flow rate–that we can
change in a precise fashion and then observe how the system behaves as
this parameter varies. These parameters make reference to structural
aspects of the systems in question, like changes in temperature
reflecting the energy input into the system.

10. So far as I know, there
is no agreement on any quantum property that could distinguish between
chaotic and non-chaotic quantum systems (Weigert 1992).

11. Wigner originally
derived these distributions for complex (heavy) nuclei by assuming
that, in the Heisenberg representation using typical basis vectors,
the matrix elements of the Hamiltonian can be treated as if they are
Gaussian random numbers. This produces a model, known as a random
matrix model, that has no free parameters and is invariant under
a wide range of change of basis. Wigner’s random matrix model was very
successful in describing the energy spectra of complex quantum
systems.

12. If the time evolution
of a system requires at least \(N\) bits of input information
about the initial state to obtain \(N\) bits of output
information about its future state, then it is algorithmically
complex. The string of information describing the future state given
the initial state cannot be compressed at all. Classical chaotic
trajectories are always algorithmically complex.

13. Even if one uses David
Bohm’s version of QM (Bohm 1951; Bohm and Hiley 1993), which has
continuous particle trajectories in spacetime, there are still
important conceptual differences (e.g., the presence of an
all-pervasive quantum potential in Bohm’s theory).

14. While it is true that
apparent indeterminism can be generated if the state space one uses to
analyze chaotic behavior is coarse-grained, this produces only an
epistemic form of indeterminism leaving the ontological character of
the underlying equations fully deterministic.

16. Many authors have
concluded that Prigogine and collaborators were arguing that
trajectories did not exist (e.g., Bricmont 1995), but this is not the
case. The matter is somewhat technical and the Brussels-Austin Group
has been notoriously unclear in writing about this point (see Bishop
2004).